Laminations are a combinatorial and topological model for studying the Julia sets of complex polynomials. Every complex polynomial of degree d has d fixed points, counted with multiplicity. From the point of view of laminations, at most d-1, of these fixed points are peripheral (approachable from outside the Julia set of the polynomial). Hence, at least one of the d fixed points is “hidden” from the laminational point of view. The purpose of this study is to identify, classify and count the possible fixed point portraits for any lamination of degree d. We will identify the “simplest” lamination for a given fixed point portrait and will show that there are polynomials that have these simplest laminations. An application of fixed point portraits is to establish a correspondence between locally unicritical laminations and locally maximally critical laminations with rotational polygons. This application is a joint work with Brittany E. Burdette.

It is well known that the set of invariant measures of a topological dynamical system is a non-empty metrisable Choquet simplex. In 1991, Downarowicz proved that all such simplices arise as the sets of invariant measures of a class of minimal subshifts. Hence, one can ask the following question: which non-empty Choquet simplices can be realised as the sets of invariant measures for minimal homeomorphisms on manifolds? In the case of one-dimensional manifolds, we observe that the geometry of manifolds restricts the available dynamics. In my talk, I will discuss which measurable dynamical system can be realised as a minimal homeomorphism on a manifold. This will answer the question of realisation of Choquet simplex on manifolds of higher dimension.

I will also talk about necessary and sufficient conditions for an ergodic measure in Choquet simplex to have a discrete spectrum. The criterion is imposed on generic points of such a measure.

The talk is based on the results obtained in joint works with Melih Emin Can, Jernej Činč, Till Hauser, Dominik Kwietniak, and Piotr Oprocha.

We use rotation theory to deduce an order among periods of mixing patterns of some maps of triods.

We study a family of infinite interval exchange transformations on the unit interval emerging from compositions of the Von Neumann-Kakutani map (dyadic odometer) with rational rotations (or more generally permutations of equal-length intervals. Hence the name ``rotated odometers''. By means of renormalization (similar to Rauzy-Veech induction) we cam translate the problem into one on symbolic substitutions, and determine the dynamic and ergodic structure of these rotated odometers.

This is joint work with Olga Lukina

This talk with discuss a method of finding Julia sets from particular laminations. We use Mathematica and Matlab to model and solve a system of equations that represent the lamination in order to find the unique corresponding Julia set. Issues surrounding this method will also be discussed.

This talk is based on a recent study of the class of interval maps with dense set of periodic points $CP$CP and its closure $\overline{CP}$CP equipped with the metric of uniform convergence where we proved the following results:

1.) $\overline{CP}$CP is dynamically characterized as the set of interval maps for which every point is chain-recurrent.

2.) Topological exactness (or leo property) is attained on the open dense set of maps in $CP$CP.

3.) Every second category set in both $CP$CP and $\overline{CP}$CP contains uncountably many conjugacy classes.

Based on a joint work with Jozef Bobok (CVUT Prague), Piotr Oprocha (AGH Krakow) and Serge Troubetzkoy (Aix Marseille).

In this talk we will discuss adic systems on Bratteli diagrams associated to multivariable polynomials. While these diagrams are not stationary, they exhibit a self-similar structure that can be used to understand any resulting adic system. In particular, the structure alone implies the diagram is inherently expansive. Further, any diagram with multivariable polynomial shape will also be inherently expansive.

A lamination L is a closed set of chords of the unit disk so that no two chords intersect in the open disk. A lamination is d-invariant under the degree d covering map $\sigma_d:S\to S$σd​:S→S of the unit circle if it is forward invariant (for any chord $ab$ab in L the chord $\sigma(a)\sigma(b)$σ(a)σ(b) is also in $L$L). In this talk we will discuss properties of d-invariant laminations that all contain a given forward invariant subset $P$P of chords (for example a given periodic chord).

We count possible preimage laminations for n steps. i.e. the number of laminations that have a particular $P$P as their $\sigma_d^n$σdn​ image. In contrast to most laminations research, we do not specify critical cords (i.e., chords $ab$ab so that $\sigma(a)=\sigma(b)$σ(a)=σ(b)).

We define what laminations should be included in our count. Particularly, we exclude critical and degenerate leaves from our laminations because they make the count immediately infinite. We also insist that each of the counted laminations are maximal, to avoid confluence, and have adequately many chords with the same image.

This class of laminations has the added advantage that they are all realized by complex polynomials of degree d, giving us some hope that we can use our combinatorial model to assemble a model of polynomial parameter space. It is clear in the degree 2 case that the laminations which we generate in our count correspond to limbs outside the molecule of the connectedness locus, with exactly one exception for each n.

Continued fractions have long been an object of interest to both number theorists and dynamicists. In the 1970's and 80's great progress was made on understanding metrical properties of continued fractions, i.e. measure-theoretic properties. A particular focus was on the the maximal digits of continued fractions and their properties. In this talk I will discuss some of these results including an extreme value law proved by Galambos and a Poisson Law by Iosifescu. I will also discuss some recent developments in the field, primarily generalisations of these results to complex continued fractions.

Amorphic complexity is a relatively new invariant of dynamical systems useful in the study of aperiodic order and low complexity dynamics. Tameness is a well-studied notion defined in terms of the size of the Ellis semigroup of the system. In the talk we will study amorphic complexity and tameness in the class of automatic systems (systems arising from constant length substitutions). We will present a closed formula for the complexity of any automatic system and show that tameness of automatic systems can be succinctly characterised using amorphic complexity: an automatic system is tame if and only if its amorphic complexity is one. The talk is based on a joint work with Maik Gröger.

Enveloping semigroups, introduced by Robert Ellis, are a useful tool in topological dynamics which allows to describe the behavior of systems (equicontinuity, distality, ...) in terms of properties of a topological-algebraic structure. Based on this idea, we discuss "enveloping semigroupoids" in this talk and how they can be used to study structured extensions in topological dynamics and ergodic theory. This is based on joint work with Nikolai Edeko, Patrick Hermle and Asgar Jamneshan.

A fundamental theme in dynamics is the classification of systems up to appropriate equivalence relations. For instance, the equivalence relation of topological conjugacy preserves the qualitative behavior of topological dynamical systems. Smale's celebrated program proposes to classify topological or smooth dynamical systems up to topological conjugacy.

These classification problems not only turn out to be hard but sometimes even to be impossible. In joint work with Deka, Garcia-Ramos, Kasprzak, and Kwietniak, we show that the equivalence relation generated by topological conjugacy of minimal homeomorphisms on a Cantor space is not a Borel set. This implies that Cantor minimal systems cannot be classified using inherently countable techniques.

The notion of the type and typeset were introduced by Baer in 1937 in order to develop a classification of rank n subgroups of $\mathbb{Q}^n$Qn. In this talk, we will introduce the notion of the type and typeset for minimal equicontinuous actions of non-abelian groups. In particular, we show that the commensurable class of the typeset is an invariant of the return equivalence class of such an action.

In this talk we will use symbolic systems developed by Baldwin to analyze the structure of inverse limits of certain unimodal maps on dendrites. In particular, we will characterize the endpoints and branchpoints of such an inverse limit in terms of the kneading sequence associated with the map.

In linear dynamics, bounded linear operators over infinite-dimensional Banach spaces have been shown to be able to exhibit interesting characteristics including topological transitivity, topological mixing, and even chaos in the sense of Devaney. This talk will examine weighted $\ell^p$ℓp sequence spaces together with the shift action as the operator. In the case the shift action is over the semi-group $\mathbb{N}$N, the above topological properties have been characterized by conditions on the weight sequence associated with a given $\ell^p$ℓp space. In this talk I will present recent results for new characterizations of these properties when we instead consider the group action over a countable group. I will also highlight other open questions.

This is a joint work with Kevin McGoff and William Brian.

In 2023 Haihambo and Olela Otafudu introduced and studied the notion of quasi-uniform entropy $h_{QU}(\psi)$hQU​(ψ) for a uniformly continuous self-map $\psi$ψ of a quasi-metric or a quasi-uniform space $X$X. In this talk, we discuss the connection between the topological entropy functions $h, h_f$h,hf​ and the quasi-uniform entropy function $h_{QU}$hQU​ on a quasi-uniform space $X$X, where $h$h and $h_f$hf​ are the topological entropy functions defined using compact sets and finite open covers, respectively. In particular, we have shown that for a uniformly continuous self-map $\psi$ψ of a $T_0$T0​-quasi-uniform space $(X,\mathcal{U})$(X,U) we have $h(\psi)\leq h_{QU}(\psi)$h(ψ)≤hQU​(ψ) when $X$X is compact and $h_{QU}(\psi)\leq h_f(\psi)$hQU​(ψ)≤hf​(ψ) with equality if $X$X is a compact $T_2$T2​ space.

Hom shifts form a class of multidimensional shifts of finite type (SFT) where adjacent symbols must be neighbors in a fixed finite undirected simple graph $G$G. This talk is about gluing distance in Hom shifts: given two $n x n$nxn admissible partial blocks, how far do they need to be so that we can glue them together (i.e embed) in a larger admissible block.

The gluing gap measures how far any two square patterns of size $n$n can be glued, which has a clear analogy with gap fo specification property in one-dimensional subshifts. We prove that the gluing gap either depends linearly on $n$n or is dominated by $log(n)$log(n). It is clear that there are Hom shift, where gluing gap is bounded by constant, thus independent of $n$n. To support our results, we find a Hom shift with gap ${\Theta}(log(n))$Θ(log(n)), infirming a conjecture formulated by R. Pavlov and M. Schraudner.

This talk is based on a joint work with Silvere Gangloff and Benjamin Hellouin de Menibus

A recent polynomial version of the celebrated Sarnak's conjecture asked whether, given a nonlinear polynomial $p \in \mathbb{Z}[x]$p∈Z[x], zero entropy minimal topological dynamical system $(X,T)$(X,T), $f \in C(X)$f∈C(X), and $x_0 \in X$x0​∈X, the sequence $f(T^{p(x)} x_0)$f(Tp(x)x0​) is uncorrelated with the Mobius function $\mu$μ.

This conjecture is false, and has been refuted in two recent works with interesting and somewhat difficult constructions. However, we can use a simple symbolic construction to prove the following: when $(k_n)$(kn​) has zero Banach density, then not only may the sequence $f(T^{k_n} x_0)$f(Tkn​x0​) be correlated with $\mu$μ, there are actually no restrictions on the sequence whatsoever.

In this talk we will study measure-theoretical rigidity and partial rigidity for classes of Cantor dynamical systems including Toeplitz systems and enumeration systems. With the use of Bratteli-Vershik dynamical systems we can control invariant measures. Their structure in the Bratteli diagram leads us to find systems with the desired properties. Among other things, we will analyse different Toeplitz systems for their rigidity and show that there exist Toeplitz systems which have zero entropy and are not partially measure theoretically rigid with respect to any of its invariant measures. Further we show varying rigidity in the family of enumeration systems defined by a linear recursion.

This talk is based on joint work with Henk Bruin, Olena Karpel and Piotr Oprocha.

The Thue-Morse measure and its generalisations are diffraction measures of simple aperiodic systems. Besides that, they are paradigmatic examples of purely singular continuous probability measures on the unit interval given as an infinite Riesz product. To study their scaling behaviour a classical method, the thermodynamic formalism can be used - which however has to be adapted to an unbounded potential. We will in particular see how one has to meaningfully define the topological and variational pressure in this setting. Besides seeing this method, we will also see how quantitatively the Birkhoff and dimension spectrum changes depending on the point of the singularity. This is joint work with M. Baake, P. Gohlke, and M. Kesseböhmer.

In this talk, we discuss the dynamics of a general non-autonomous dynamical system. In particular, we discuss notions like equicontinuity, minimality and various notions of mixing and sensitivities for a general discrete non-autonomous system. We also discuss the case when the dynamics is generated by a uniformly convergent sequence of maps. We prove that if the system is generated by a commutative family converging at a "sufficiently fast rate" then many dynamical notions for non-autonomous system can be characterized by the limiting (autonomous) system.

Every complex polynomial with a locally connected Julia set generates a lamination of the unit disc --- a closed set of non-crossing chords inside $\mathbb{D}$D whose endpoints on $\mathbb{S}$S are allowed to touch. This was a tool developed and explored by Thurston in order to study degree 2 complex polynomials, their Julia sets, and the parameter space of LC polynomials, the Mandelbrot set. The dimension of the corresponding `multi-brot' sets increases in higher dimension so one usually restricts themselves to studying slices of the parameter space.

The restriction we make in this talk is to focus on symmetric polynomials, which we define as a degree $d$d complex polynomial whose locally connected Julia set --- and therefore, whose lamination --- has $\frac{2\pi}{d-1}$d−12π​ rotational symmetry. It turns out that this leads to behavior very similar to the degree 2 case.

Thurston utilized the Central Strip Lemma to help prove two main results in the degree 2 case --- the No Wandering Triangles theorem (NWT) and the No Identity Return Triangles theorem (NIRT). The symmetric degree $d$d case has an analogous result, which we call the Critical Strip Lemma. In this talk we prove the Critical Strip Lemma, which puts restrictions on the placement of leaves in $\sigma_d$σd​-symmetric laminations. We will then outline how it's used to prove that in the $\sigma_d$σd​-symmetric case, the NWT and NIRT theorems still hold.

During the talk we will focus on surjective Cantor systems. Each such system can be easily embedded in the Gehman dendrite, as its set of endpoints is a Cantor set. We will show that for each such embedding there exists a mixing map of the dendrite such that the endpoints' subsystem is conjugate to the Cantor system of choice. The main tool to obtain this result follows from Shimomura's method of approximating the dynamics on zero dimensional systems by analysing the dynamics of coverings of the underlying space. We will discuss the dynamical properties of the constructed map.

The talk is based on joint work with Dominik Kwietniak and Piotr Oprocha.

In this talk, we discuss the topological dynamics of a general non autonomous dynamical system. In particular, we discuss various dynamical properties such as equicontinuity, minimality, almost periodicity, proximality. We introduce the notion of orbital hull in non-autonomous dynamical systems and relate it with the dynamics of the system. We give a necessary and sufficient condition for the system to be minimal in terms of orbital hull. We also relate almost periodicity of the orbital hull of a point to equicontinuity of the non-autonomous dynamical system. We show that any minimal system generated by commutative family is either equicontinuous or has a dense set of sensitive points. We also discuss weakly mixing and proximality fo non-autonomous systems. We generalize some results for autonomous system to non-autonomous setting. We also discuss some counter-examples for the case when result cannot be extended to a non-autonomous setting.